API-201 Final Exam
QUESTION 1: AGRICULTURAL UNCERTAINTY
When a particular farmer chooses to grow a crop, assume that he may end up with too
little rainfall (30% of the time), too much rainfall (20% of the time), or just the right
amount (50% of the time). Assume that he also may suffer a serious pest infestation (10%
of the time). Assume that rainfall and pest infestations are independent.
Answer the following questions:
(a) (4 points) What is the probability that the farmer has the right amount of rainfall
and no pest problem?
(b) (4 points) If you know that the farmer had a pest problem, what is the probability that he got too much rain?
(c) (4 points) What is the probability that the farmer has a pest problem or too much rain (or both)?
Assume that the farmer earns a profit of 10,000 rupees if the rain is just right and there is no pest infestation; in all other cases, his profit is 0 rupees. Instead of farming the crop, the farmer can do manual labor on a local road project. If he does manual labor, he will earn 4,200 rupees. (d) (8 points) If he only cares about his earnings and is risk-neutral, should the farmer farm the crop or do manual labor? Draw a decision tree to illustrate.
(e) (6 points) If the farmer is risk-averse, would this change his decision in part (d)? Briefly explain.
(10 points) Rather than planting normal seeds, the farmer could plant droughtresistant seeds. These seeds allow the farmer to earn a profit of 10,000 rupees both when the rain is just right and when it is too little (so long as there is no pest infestation, since no seeds protect against pests). What is the most the farmer would be willing to pay for the drought resistant seeds assuming that he is risk neutral?
Now assume that pests require water and so there is never a pest infestation when there is too little rain. Otherwise, when there is just enough rain, there is a 10% chance of a pest infestation; when there is too much rain, there is a 25% chance of a pest infestation. (g) (6 points) Now if you know that the farmer had a pest problem, what is the probability that he got too much rain?
QUESTION 2: RECIDIVISM IN CALIFORNIA “Recidivism” is when a convicted criminal re-offends. In other words, a criminal is jailed for a crime, released, and then winds up in jail again later. Out of all U.S. states, California has the highest recidivism rate for convicted criminals: overall, 65% of criminals released from jail end up back in jail within three years, a rate substantially higher than most other states. A government statistician has calculated the average cost of each case of recidivism to be $400,000 (considering all costs to society, all of which the State of California considers in its decisions). Say that you are the governor of California and you are reviewing a report on an innovative new program to reduce recidivism in another state. The report details a randomized experiment used to evaluate the new program. It found an average causal effect of -10 percentage points: among their study population, 25% of those in the treatment group re-offended within three years, vs. 35% of those in the control group. Answer the following questions: (a) (6 points) If you were to implement this program in California, would you expect a 10 percentage-point reduction in the recidivism rate? Briefly explain (as if to your statistics professor) why you would or wouldn’t, and be specific about what other information you might want (if any).
Now assume that, to implement the program in California, it would cost you $30,000 per criminal put through the program. Assume also that the program would have a -10 percentage-point effect on recidivism. (b) (8 points) Next month, 1000 criminals will be released. Would you choose to implement the program for this group? Illustrate your decision with a decision tree.
(c) (6 points) Suppose your budget only allowed you to implement the program for 30% of released criminals. What is the expected recidivism rate assuming that you implement the program for a random 30% subset of released criminals?
When pressed, the government statisticians who estimated the $400,000/case cost of recidivism admitted that the cost is very different for criminals in jail for violent vs. nonviolent crimes. Each case of recidivism for a violent criminal costs $1,000,000, while for a non-violent criminal it costs only $250,000. Only 20% of criminals are violent. (d) (12 points) Assume that the recidivism rate and program effectiveness are both independent of whether the criminal is violent or non-violent. For the 1000 criminals to be released next month, would you implement the program for all of them, just the violent criminals, or for none of them? Again, illustrate with a decision tree.
(e) (4 points) Do you think that it is likely to be a valid assumption that recidivism rate and whether a criminal is violent or non-violent are independent? Why or why not? Explain briefly.
(f) (6 points) If this assumption of independence is not valid, which elements of your decision tree would change? Briefly explain which elements would change, how they would change, and whether they would increase or decrease any expected payoffs you calculated.
QUESTION 3: RECIDIVISM IN CALIFORNIA, PART II We continue with the problem of recidivism in California. You decide to commission a study to estimate the recidivism rate for violent criminals. (a) (5 points) Assuming a significance level of 0.05, what is the largest margin of error you would expect from a random sample of 900 violent criminals?
(b) (5 points) What sample size would you need to have a margin of error no larger than plus or minus 2 percentage points (again assuming a significance level of 0.05)?
While the researchers were busy trying to draw a random sample for this study, a hardworking department head found the following data for the full population of violent criminals released 3 years ago.
(c) (4 points) What does this table imply about the correlation between recidivism and age? Briefly explain.
(d) (4 points) Based on the table above, is it likely that either the mean age or the median age of released violent offenders is higher? Briefly explain.
QUESTION 4 – TRUE/FALSE/UNCERTAIN and EXPLAIN (a) (6 points) If you change the significance level α for a hypothesis test so that it is closer to 0, then you reduce both the probability of falsely rejecting the null hypothesis when it is true and correctly rejecting the null hypothesis when it is false.
(b) (6 points) A researcher is planning to conduct a survey of amateur athletes in 24 countries. Using the data, she intends to analyze how the responses of Olympic athletes might differ from those of other amateur athletes. Therefore, she should use a clustered sampling design.
QUESTION 5 – TRUE/FALSE/UNCERTAIN and EXPLAIN (6 points) An expert reports that, if Greece stays in the Euro zone, then the probability that Spain leaves the Euro zone is 0. If that is true, then the probability that Greece stays in the Euro zone given that Spain leaves the Euro zone is also 0.
QUESTION 6: FLOODING FROM HURICANE SANDY High waves from Hurricane Sandy flooded many of the coastal communities in New Jersey, Maryland, and Connecticut. The depths of the flood waters differed by local geography, which magnified or dampened the storm surge. (a) (8 points) Prior to the storm, Chris Christie, the governor of New Jersey, must choose whether to issue a mandatory evacuation order for Atlantic City. The costs and benefits of evacuation depend on whether Hurricane Sandy will cause extensive damage to Atlantic City. If the hurricane causes extensive damage, the cost is $100 million if the city is not evacuated and $40 million if the city is evacuated. If the hurricane does not cause extensive damage, the cost is $0 if the city is not evacuated and $70 million if the city is evacuated. At the time, the National Weather Service estimated that Hurricane Sandy would cause extensive damage to Atlantic City with 60% probability. What decision should Chris Christie make? Illustrate your decision with a decision tree.
(b) (4 points) If you know that the farmer had a pest problem, what is the probability that he got too much rain?
(c) (4 points) What is the probability that the farmer has a pest problem or too much rain (or both)?
Assume that the farmer earns a profit of 10,000 rupees if the rain is just right and there is no pest infestation; in all other cases, his profit is 0 rupees. Instead of farming the crop, the farmer can do manual labor on a local road project. If he does manual labor, he will earn 4,200 rupees. (d) (8 points) If he only cares about his earnings and is risk-neutral, should the farmer farm the crop or do manual labor? Draw a decision tree to illustrate.
(e) (6 points) If the farmer is risk-averse, would this change his decision in part (d)? Briefly explain.
(10 points) Rather than planting normal seeds, the farmer could plant droughtresistant seeds. These seeds allow the farmer to earn a profit of 10,000 rupees both when the rain is just right and when it is too little (so long as there is no pest infestation, since no seeds protect against pests). What is the most the farmer would be willing to pay for the drought resistant seeds assuming that he is risk neutral?
Now assume that pests require water and so there is never a pest infestation when there is too little rain. Otherwise, when there is just enough rain, there is a 10% chance of a pest infestation; when there is too much rain, there is a 25% chance of a pest infestation. (g) (6 points) Now if you know that the farmer had a pest problem, what is the probability that he got too much rain?
QUESTION 2: RECIDIVISM IN CALIFORNIA “Recidivism” is when a convicted criminal re-offends. In other words, a criminal is jailed for a crime, released, and then winds up in jail again later. Out of all U.S. states, California has the highest recidivism rate for convicted criminals: overall, 65% of criminals released from jail end up back in jail within three years, a rate substantially higher than most other states. A government statistician has calculated the average cost of each case of recidivism to be $400,000 (considering all costs to society, all of which the State of California considers in its decisions). Say that you are the governor of California and you are reviewing a report on an innovative new program to reduce recidivism in another state. The report details a randomized experiment used to evaluate the new program. It found an average causal effect of -10 percentage points: among their study population, 25% of those in the treatment group re-offended within three years, vs. 35% of those in the control group. Answer the following questions: (a) (6 points) If you were to implement this program in California, would you expect a 10 percentage-point reduction in the recidivism rate? Briefly explain (as if to your statistics professor) why you would or wouldn’t, and be specific about what other information you might want (if any).
Now assume that, to implement the program in California, it would cost you $30,000 per criminal put through the program. Assume also that the program would have a -10 percentage-point effect on recidivism. (b) (8 points) Next month, 1000 criminals will be released. Would you choose to implement the program for this group? Illustrate your decision with a decision tree.
(c) (6 points) Suppose your budget only allowed you to implement the program for 30% of released criminals. What is the expected recidivism rate assuming that you implement the program for a random 30% subset of released criminals?
When pressed, the government statisticians who estimated the $400,000/case cost of recidivism admitted that the cost is very different for criminals in jail for violent vs. nonviolent crimes. Each case of recidivism for a violent criminal costs $1,000,000, while for a non-violent criminal it costs only $250,000. Only 20% of criminals are violent. (d) (12 points) Assume that the recidivism rate and program effectiveness are both independent of whether the criminal is violent or non-violent. For the 1000 criminals to be released next month, would you implement the program for all of them, just the violent criminals, or for none of them? Again, illustrate with a decision tree.
(e) (4 points) Do you think that it is likely to be a valid assumption that recidivism rate and whether a criminal is violent or non-violent are independent? Why or why not? Explain briefly.
(f) (6 points) If this assumption of independence is not valid, which elements of your decision tree would change? Briefly explain which elements would change, how they would change, and whether they would increase or decrease any expected payoffs you calculated.
QUESTION 3: RECIDIVISM IN CALIFORNIA, PART II We continue with the problem of recidivism in California. You decide to commission a study to estimate the recidivism rate for violent criminals. (a) (5 points) Assuming a significance level of 0.05, what is the largest margin of error you would expect from a random sample of 900 violent criminals?
(b) (5 points) What sample size would you need to have a margin of error no larger than plus or minus 2 percentage points (again assuming a significance level of 0.05)?
While the researchers were busy trying to draw a random sample for this study, a hardworking department head found the following data for the full population of violent criminals released 3 years ago.
(c) (4 points) What does this table imply about the correlation between recidivism and age? Briefly explain.
(d) (4 points) Based on the table above, is it likely that either the mean age or the median age of released violent offenders is higher? Briefly explain.
QUESTION 4 – TRUE/FALSE/UNCERTAIN and EXPLAIN (a) (6 points) If you change the significance level α for a hypothesis test so that it is closer to 0, then you reduce both the probability of falsely rejecting the null hypothesis when it is true and correctly rejecting the null hypothesis when it is false.
(b) (6 points) A researcher is planning to conduct a survey of amateur athletes in 24 countries. Using the data, she intends to analyze how the responses of Olympic athletes might differ from those of other amateur athletes. Therefore, she should use a clustered sampling design.
QUESTION 5 – TRUE/FALSE/UNCERTAIN and EXPLAIN (6 points) An expert reports that, if Greece stays in the Euro zone, then the probability that Spain leaves the Euro zone is 0. If that is true, then the probability that Greece stays in the Euro zone given that Spain leaves the Euro zone is also 0.
QUESTION 6: FLOODING FROM HURICANE SANDY High waves from Hurricane Sandy flooded many of the coastal communities in New Jersey, Maryland, and Connecticut. The depths of the flood waters differed by local geography, which magnified or dampened the storm surge. (a) (8 points) Prior to the storm, Chris Christie, the governor of New Jersey, must choose whether to issue a mandatory evacuation order for Atlantic City. The costs and benefits of evacuation depend on whether Hurricane Sandy will cause extensive damage to Atlantic City. If the hurricane causes extensive damage, the cost is $100 million if the city is not evacuated and $40 million if the city is evacuated. If the hurricane does not cause extensive damage, the cost is $0 if the city is not evacuated and $70 million if the city is evacuated. At the time, the National Weather Service estimated that Hurricane Sandy would cause extensive damage to Atlantic City with 60% probability. What decision should Chris Christie make? Illustrate your decision with a decision tree.
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